Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvat | Structured version Visualization version GIF version | ||
| Description: If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 32175 analog.) (Contributed by NM, 11-Jan-2015.) |
| Ref | Expression |
|---|---|
| lcvat.s | β’ π = (LSubSpβπ) |
| lcvat.p | β’ β = (LSSumβπ) |
| lcvat.a | β’ π΄ = (LSAtomsβπ) |
| icvat.c | β’ πΆ = ( βL βπ) |
| lcvat.w | β’ (π β π β LMod) |
| lcvat.t | β’ (π β π β π) |
| lcvat.u | β’ (π β π β π) |
| lcvat.l | β’ (π β ππΆπ) |
| Ref | Expression |
|---|---|
| lcvat | β’ (π β βπ β π΄ (π β π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcvat.s | . . 3 β’ π = (LSubSpβπ) | |
| 2 | lcvat.p | . . 3 β’ β = (LSSumβπ) | |
| 3 | lcvat.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
| 4 | lcvat.w | . . 3 β’ (π β π β LMod) | |
| 5 | lcvat.t | . . 3 β’ (π β π β π) | |
| 6 | lcvat.u | . . 3 β’ (π β π β π) | |
| 7 | icvat.c | . . . 4 β’ πΆ = ( βL βπ) | |
| 8 | lcvat.l | . . . 4 β’ (π β ππΆπ) | |
| 9 | 1, 7, 4, 5, 6, 8 | lcvpss 38496 | . . 3 β’ (π β π β π) |
| 10 | 1, 2, 3, 4, 5, 6, 9 | lrelat 38486 | . 2 β’ (π β βπ β π΄ (π β (π β π) β§ (π β π) β π)) |
| 11 | 4 | 3ad2ant1 1131 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β LMod) |
| 12 | 5 | 3ad2ant1 1131 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 13 | 6 | 3ad2ant1 1131 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 14 | simp2 1135 | . . . . . . 7 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π΄) | |
| 15 | 1, 3, 11, 14 | lsatlssel 38469 | . . . . . 6 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 16 | 1, 2 | lsmcl 20967 | . . . . . 6 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
| 17 | 11, 12, 15, 16 | syl3anc 1369 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) β π) |
| 18 | 8 | 3ad2ant1 1131 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β ππΆπ) |
| 19 | simp3l 1199 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β (π β π)) | |
| 20 | simp3r 1200 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) β π) | |
| 21 | 1, 7, 11, 12, 13, 17, 18, 19, 20 | lcvnbtwn2 38499 | . . . 4 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) = π) |
| 22 | 21 | 3exp 1117 | . . 3 β’ (π β (π β π΄ β ((π β (π β π) β§ (π β π) β π) β (π β π) = π))) |
| 23 | 22 | reximdvai 3162 | . 2 β’ (π β (βπ β π΄ (π β (π β π) β§ (π β π) β π) β βπ β π΄ (π β π) = π)) |
| 24 | 10, 23 | mpd 15 | 1 β’ (π β βπ β π΄ (π β π) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwrex 3067 β wss 3947 β wpss 3948 class class class wbr 5148 βcfv 6548 (class class class)co 7420 LSSumclsm 19588 LModclmod 20742 LSubSpclss 20814 LSAtomsclsa 38446 βL clcv 38490 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-lsm 19590 df-cmn 19736 df-abl 19737 df-mgp 20074 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lsatoms 38448 df-lcv 38491 |
| This theorem is referenced by: islshpcv 38525 |
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